Properties

Label 2-84e2-1.1-c1-0-66
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 5·11-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s − 4·25-s − 3·29-s − 5·31-s − 2·37-s + 8·41-s + 4·43-s − 4·47-s + 9·53-s + 5·55-s + 3·59-s − 12·61-s − 2·65-s − 2·67-s + 8·71-s − 14·73-s − 79-s − 17·83-s − 6·85-s + 18·89-s + 2·95-s + 3·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.557·29-s − 0.898·31-s − 0.328·37-s + 1.24·41-s + 0.609·43-s − 0.583·47-s + 1.23·53-s + 0.674·55-s + 0.390·59-s − 1.53·61-s − 0.248·65-s − 0.244·67-s + 0.949·71-s − 1.63·73-s − 0.112·79-s − 1.86·83-s − 0.650·85-s + 1.90·89-s + 0.205·95-s + 0.304·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.56961595165383949766549509354, −7.18258949286253915460328854513, −5.97187493805965915604094039402, −5.55831683575957568269547090351, −4.79945078049509759639910661580, −3.87417469111949783205094110036, −3.19326327725757242519286026200, −2.37825403551307500139998559765, −1.20985222033741554411862467683, 0, 1.20985222033741554411862467683, 2.37825403551307500139998559765, 3.19326327725757242519286026200, 3.87417469111949783205094110036, 4.79945078049509759639910661580, 5.55831683575957568269547090351, 5.97187493805965915604094039402, 7.18258949286253915460328854513, 7.56961595165383949766549509354

Graph of the $Z$-function along the critical line